Decision Problems for Probabilistic Finite Automata on Bounded Languages

We show that several problems concerning probabilistic finite automata of a fixed dimension and a fixed number of letters for bounded cut-point and strict cut-point languages are algorithmically undecidable by a reduction of Hilbert’s tenth problem. We then consider the set of so called “F-Problems” (emptiness, infiniteness, containment, disjointness, universe and equivalence) and show that they are also undecidable for bounded (non-)strict cut-point languages on probabilistic finite automata. For a finite set of matrices { M1 , M2 , . . . , Mk }⊆ Q txt , we then consider the decidability of computing the maximal spectral radius of any matrix in the set X = { Mj11 Mj22··· Mjkk|j1, j2, . . . , jk≥0 } , which we call a bounded matrix language. Using an encoding of a probabilistic finite automaton shown in the paper, we prove the surprising result that determining if the maximal spectral radius of a bounded matrix language is less than or equal to one is undecidable, but determining whether it is strictly less than one is in fact decidable (which is similar to a result recently shown for quantum automata).